Mathematics for the interested outsider

Hom-Space Additivity

Today I’d like to show that the space of homomorphisms between two -modules is “additive”. That is, it satisfies the isomorphisms:

We should be careful here: the direct sums inside the are direct sums of -modules, while those outside are direct sums of vector spaces.

The second of these is actually the easier. If is a -morphism, then we can write it as , where and . Indeed, just take the projection and compose it with to get . These projections are also -morphisms, since and are -submodules. Since every can be uniquely decomposed, we get a linear map .

Then the general rules of direct sums tell us we can inject and back into , and write

Thus given any -morphisms and we can reconstruct an . This gives us a map in the other direction — — which is clearly the inverse of the first one, and thus establishes our isomorphism.

Now that we’ve established the second isomorphism, the first becomes clearer. Given a -morphism we need to find morphisms . Before we composed with projections, so this time let’s compose with injections! Indeed, composes with to give . On the other hand, given morphisms , we can use the projections and compose them with the to get two morphisms . Adding them together gives a single morphism, and if the came from an , then this reconstructs the original. Indeed:

And so the first isomorphism holds as well.

We should note that these are not just isomorphisms, but “natural” isomorphisms. That the construction is a functor is clear, and it’s straightforward to verify that these isomorphisms are natural for those who are interested in the category-theoretic details.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.